
Notice that while the above assumption is restrictive, there are practical scenarios where one can expect it to hold. Furthermore, the assumption has been studied in prior works to understand noise sensitivity of post-processing and in-processing methods for building fair classifiers [1]}, [2]}. Two settings where such an assumption would hold are as follows. First, this would hold if the attribute classifier uses a set of features that are conditionally independent of the features used by the label classifier. The assumption will also hold if the attribute classifier makes independent errors with a certain probability. This, for instance, might be the case for certain classifiers based upon crowdsourcing. As we will soon see, even under the above simplified setting, the structure of the optimal attribute classifier can defy convention wisdom. Using the attribute classifier \(h\) , we get estimates \(\hat{\alpha }, \hat{\beta }\)  of the true values \(\alpha , \beta \)  (see Eq. (REF )), where
\(\hat{\alpha } &= \mathbb {P} (f(x)=1 | y=1, \hat{a}=1),\\\hat{\beta } &= \mathbb {P} (f(x)=1 | y=1, \hat{a}=0).\) 
